Ideals Of Lie Algebras Research Articles

Inner ideals in Lie algebras and spherical buildings

The correspondence found by Faulkner between inner ideals of the Lie algebra of a simple algebraic group and shadows on the set of long root groups of the building associated with the algebraic group is shown to hold in greater generality (in particular, over fields of characteristic distinct from two).

On ϕ-ideals and the structure of Lie algebras

This paper aims to study the concept of [Formula: see text]-ideals of a finite-dimensional Lie algebra which is analogous to the concept of [Formula: see text]-ideal and [Formula: see text]-normal subgroup. We compile some basic properties of [Formula: see text]-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for [Formula: see text]-ideals of Lie algebras and [Formula: see text]-normal subgroups are not valid here.

Foliated groupoids and infinitesimal ideal systems

The main goal of this work is to introduce a natural notion of ideal in a Lie algebroid, the “infinitesimal ideal systems”. Ideals in Lie algebras and the Bott connection associated to involutive subbundles of tangent bundles are special cases. The definition of these objects is motivated by the infinitesimal description of involutive multiplicative distributions on Lie groupoids. In the Lie group case, such distributions correspond to ideals. Several examples of infinitesimal ideal systems are presented, and (under suitable regularity conditions) the quotient of a Lie algebroid by an infinitesimal ideal system is shown to be a Lie algebroid.

An Elemental Characterization of Orthogonal Ideals in Lie Algebras

In this paper, we prove that the ideals generated by two elements x, y in a nondegenerate Lie algebra L over a ring of scalars Φ with \({\frac 1 2, \frac 1 3}\) are orthogonal if and only if [x, [y, L]] = 0.

Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finite-dimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. Finally, we determine the maximal dimension of abelian subalgebras and ideals for non-decomposable solvable non-nilpotent Lie algebras of dimension 6 over both the fields ℝ and ℂ, showing the differences between these fields.

On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log2 p.

Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras

Chain conditions for abelian, nilpotent and soluble ideals in Lie algebras

Maximal conditions for ideals in Lie algebras

Maximal conditions for ideals in Lie algebras

Finiteness conditions for Abelian ideals and nilpotent ideals in Lie algebras

Finiteness conditions for Abelian ideals and nilpotent ideals in Lie algebras

The Lie inner ideal structure of associative rings

An inner ideal of an associative ring R is an additive subgroup V of R such that [V[ VR]] C I/. This paper examines the inner ideal structure of semiprime associative rings and of the skew elements of prime rings with involution. The results are analogous to those obtained by Herstein, Baxter, and Erickson for the Lie ideals of these rings. In the special case when R is a simple Artinian ring with center 2 of characteristic not 2 or 3, and of dimension greater than 16 over 2, then [RR]/2 n [RR] is a simple Lie algebra over Z satisfying both the ascending and descending chain conditions on inner ideals. Every inner ideal has the form eRf for e, f idempotents of R such that fe = 0. Moreover, if * is an involution on R and if K denotes the skew elements relative to *, then [AX]/2 n [KK] also satisfies both chain conditions on inner ideals. Every inner ideal of this algebra can be written as eKe* for some idempotent e of R such that e*e = 0. The motivation to study inner ideals in Lie algebras can be found in a recent paper [3] by the author. The inner ideals of a Lie algebra are closely related to the ad-nilpotent elements, and certain restrictions on the adnilpotent elements yield an elementary criterion for distinguishing the nonclassical from classical simple Lie algebras over algebraically closed fields of characteristic p > 5. In what follows R is a noncommutative, associative ring. Say R is n-torsion free where n is a positive integer if, in R, nx = 0 implies x = 0. Let 2 denote the center of R, and given elements a, b in R, let [ab] = ab ~ ba. Then 2 {a E R 1 [ab] = 0 for all b in Ii}. Alternatively, given any a in R if D,,(r) = [ar], then 2 = {a E R / D, = O}. For subsets A, B of R use [AB]

Densely embedded ideals of lie algebras

Densely embedded ideals of lie algebras

On prime ideals of Lie algebras

On prime ideals of Lie algebras